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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B / d / Solution

Contracting the Riemann curvature tensor gives the nonzero components of the Ricci tensor:

R_{rr} = R^{ϕ}_{r ϕ r} = - 1, R_{ϕϕ} = R^{r}_{ϕ r ϕ} = - sinh^{2} r .

The Ricci scalar is the contraction with the inverse metric. Therefore

R = g^{rr} R_{rr} + g^{ϕϕ} R_{ϕϕ} = \frac{1}{a ^{2}} (- 1) + \frac{1}{a ^{2} sinh ^{2} r} (- sinh^{2} r) = - \frac{2}{a ^{2}} .

It is independent of both coordinates, so the hyperbolic plane has constant negative scalar curvature.

Solved by gpt-5.6-sol high.

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